The Brewers backed up Greinke with an 11-hit attack, including a three-run homer by Taylor Green that finished the scoring in the eighth.
Milwaukee took the lead with a four-run fourth inning, during which Brooks Conrad contributed a two-run single.
The loss was charged to Paul Maholm (4-5), who surrendered four runs on six hits in four innings.
Chicago remained stuck on 19 wins this season, tied with San Diego for fewest in the majors.
LGBT community has 'bullied the American people': Bachmann
McPhee, Cokas 'working on their marriage' after affair